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Detecting and creating oscillations using multifractal methods
Author(s) -
Seuret Stéphane
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510417
Subject(s) - multifractal system , mathematics , gravitational singularity , series (stratigraphy) , spectrum (functional analysis) , function (biology) , context (archaeology) , operator (biology) , wavelet , mathematical analysis , pure mathematics , fractal , mathematical physics , physics , quantum mechanics , computer science , biochemistry , evolutionary biology , artificial intelligence , gene , transcription factor , biology , paleontology , chemistry , repressor
By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f , we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ∑ j ∑ kd j,k ψ j,k ∈ L 2 (ℝ) the function series f t whose wavelet coefficients are d tj,k = d j,k 1 | d j,k|≥2 – jγ, for some fixed real number γ > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series f t may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f ∈ L 2 (ℝ) such that the associated thresholded function series f t effectively possesses oscillating singularities which were not present in the initial function f . This series f t is a typical example of function with homogeneous non‐concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)